Knowledge About How to Teach Mathematics

It is not just what mathematics teachers know, but how they know it and what they are 

able to mobilize mathematically in the course of teaching.

—Ball (2000, p. 243)

This knowledge is explicitly multidimensional.

—Hill, Ball, & Schilling (2008, p. 396)

The notion of a specialized knowledge base for teaching has been in existence for more than twenty years. In 1986 Lee Shulman identified a specialized form of teacher knowledge necessary for the practice of effective teaching: pedagogical content knowledge. Shulman defined pedagogical content knowledge as the knowledge and means of “representing and formulating the subject that make it comprehensible to others” (Shulman, 1986, p. 9). Shulman argued that content-absent pedagogy is problematic to the classroom teacher, who relies on content knowledge to deliver instruction and advance student learning.

Knowledge about how to teach mathematics differs in important ways from content knowledge possessed by professionals in other mathematics-related disciplines (Hill, Ball, & Schilling, 2008). Mathematics teachers must know not only the content they teach, but also how students’ knowledge of mathematics is developed and structured; how to manage internal and external representations of mathematical concepts; how to make students’ understanding of mathematics visible; and how to diagnose student misunderstandings and misconceptions, correct them, and guide them in reconstructing complex conceptual knowledge of mathematics (Ball, Lubienski, & Mewborn, 2001; Cohen & Hill, 2000; Darling-Hammond, 1999; Fennema & Franke, 1992). Moreover, teachers must understand how students reason and employ strategies for solving mathematical problems and how students apply or generalize problem-solving methods to various mathematical contexts (Cobb, 1986). The use of language, construction of metaphors and scenarios appropriate to teaching mathematical concepts, and understandings and use of curricular resources in the practice of teaching constitute a knowledge base for teaching that is specific to and grounded in the teaching of mathematics. These understandings represent the “specialized content knowledge” (Hill & Ball, 2004, p. 333) effective teachers of mathematics possess (Hill, Ball, & Schilling, 2008; Hill, Rowan, & Ball, 2005). 

Moreover, this knowledge transcends traditional subject-matter knowledge—or “common knowledge of content  and knowledge of classroom practices for teaching mathematics. Rather, knowledge of mathematics for teaching is embedded in the practice of teaching mathematics: in mathematics, how teachers hold knowledge may matter more than how much knowledge they hold. In other words, teaching quality might not relate so much to performance on standard tests of mathematics achievement as it does to whether teachers’ knowledge is procedural or conceptual, whether it is connected to big ideas or small bits, or whether it is compressed or conceptually unpacked.…student learning might result not only from teachers’ content knowledge but also from the interplay between teachers’ knowledge of students, their learning, and strategies for improving that learning. (Hill & Ball, 2004, p. 332)

Refinements in conceptualizing knowledge of mathematics for teaching are currently underway. Hill and her colleagues have developed a conception of “knowledge of content and students” (Hill, Ball, & Schilling, 2008), which is one component of pedagogical content knowledge yet differs from “knowledge of content and teaching” and “knowledge of curriculum” (Hill, Ball, & Schilling, 2008, p. 377).

Although this concept requires further development and clearer investigative results, it is critical that teacher educators and professional development providers keep students and their learning at the center rather than periphery and focus deliberately on student learning processes and outcomes when designing teacher learning activities (Messick, 2005). Beginning with what mathematics students know and our understandings of how they know it is an effective approach to bolstering the quality of instruction and student learning (Carpenter & Fennema, 1992; Carpenter, Fennema & Franke, 1996).